Scheduling Start-up Process for Time-constrained Single-arm Cluster Tools

ABSTRACT

Due to the trend of using larger wafer diameter and smaller lot size, cluster tools need to switch from processing one lot of wafers to another frequently. It leads to more transient periods in wafer fabrication. Their efficient scheduling and control problems become more and more important. It becomes difficult to solve such problems, especially when wafer residency time constraints must be considered. This work develops a Petri net model to describe the behavior during the start-up transient processes of a single-arm cluster tool. Then, based on the model, for the case that the difference of workloads among the steps is not too large and can be properly balanced, a scheduling algorithm to find an optimal feasible schedule for the start-up process is given. For other cases schedulable at the steady state, a linear programming model is developed to find an optimal feasible schedule for the start-up process.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Patent Application No. 62/221,034, filed on Sep. 20, 2015, which is incorporated by reference herein in its entirety.

LIST OF ABBREVIATIONS

CP control policy

LL loadlock

LPM linear programming model

PM process module

PN Petri net

BACKGROUND

1. Field of the invention

The present invention generally relates to scheduling a cluster tool, where the cluster tool has a single-arm robot for wafer handling, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint. In particular, the present invention relates to a method for scheduling a start-up process for a single-arm cluster tool with wafer residency time constraints.

LIST OF REFERENCES

There follows a list of references that are occasionally cited in the specification. Each of the disclosures of these references is incorporated by reference herein in its entirety.

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2. Description of the Related Art

In semiconductor manufacturing, wafers are processed in cluster tools with a single-wafer processing technology. Such technology allows manufacturers to process wafers one by one at each process module (PM) in cluster tools. These tools can provide a reconfigurable, flexible and efficient environment, leading to better quality control and reduced lead time [Bader et al., 1990; and Burggraaf, 1995]. In a cluster tool, there are several process modules (PMs), an aligner, a wafer handling robot, and loadlocks (LLs) for wafer cassette loading/unloading. All these modules are mechanically linked together in a radial way and computer-controlled. The robot in the center of the tool can have a single arm or dual arms, thus resulting in a single- or a dual-arm cluster tool as respectively shown in FIGS. 1A and 1B.

With two LLs, a cluster tool can be operated consecutively without being interrupted such that it can operate in a steady state for most of time. Great efforts have been made in its modeling and performance evaluation [Chan et al., 2011; Ding et al., 2006; Perkinston et al., 1994; Perkinston et al., 1996; Venkatesh et al., 1997; Wu and Zhou, 2010a; Yi et al., 2008; Zuberek, 2001; and Lee et al., 2014]. It is found that, under the steady state, a cluster tool operates in two different regions: transport and process-bound ones. For the former, its robot is always busy and the robot task time in a cycle determines its cycle time; while for the latter, its robot has idle time in a robot task cycle and thus the processing time of its PMs dominates its cycle time. Since the robot moving time from one PM to another is much shorter than wafer processing time [Kim et al., 2003], a backward scheduling is optimal for single-arm cluster tools [Lee et al., 2004; and Lopez and Wood, 2003]. For a dual-arm cluster tool, a swap strategy is efficient [Venkatesh et al., 1997] for it can simplify robot tasks and thus reduces cycle time.

For some wafer fabrication processes, a strict constraint on the wafer sojourn time in a PM called residency time constraint must be considered in scheduling a cluster tool [Kim et al., 2003; Lee and Park, 2005; Rostami et al., 2001; and Yoon and Lee, 2005]. Such a constraint requires that a wafer should be unloaded from a PM within a limited time after being processed; otherwise, the wafer would be damaged due to the high temperature and residual chemical gas in the PM. However, no buffer between PMs in a cluster tool makes it complicated to schedule the tool to satisfy wafer residency time constraints. Methods are presented in [Kim et al., 2003; Lee and Park, 2005; and Rostami et al., 2001] to solve this scheduling problem and find an optimal periodic schedule for dual-arm cluster tools. Necessary and sufficient schedulability conditions are proposed for both single- and dual-arm cluster tools and if schedulable, closed-form scheduling algorithms are derived to find the optimal cyclic schedules [Wu et al., 2008a; and Wu and Zhou, 2010b].

Due to the trends of larger wafer diameter and smaller lot sizes, cluster tools need to switch from processing one lot of wafers to another one frequently. This leads to more transient periods in wafer fabrication, which includes start-up and close-down processes. Their efficient scheduling and control problems become more and more important. They become very difficult to solve especially when wafer residency time constraints must be considered. Although most existing studies [Chan et al., 2011; Ding et al., 2006; Perkinston et al., 1994; Perkinston et al., 1996; Venkatesh et al., 1997; Wu and Zhou, 2010a; Yi et al., 2008; Zuberek, 2001; Qiao et al., 2012a and 2012b; Qiao et al., 2013; and Lee et al., 2014] aim at finding an optimal periodical schedule, few researches focus on scheduling for transient states [Lee et al., 2012 and 2013; Kim et al., 2012, 2013a, 2013b, and 2013c; and Wikborg and Lee, 2013] despite their increasing importance. In [Kim et al., 2012], with a given robot task sequence, the transient period for the start-up and close-down processes is minimized for a dual-arm cluster tool. In [Kim et al., 2013a, and Wikborg and Lee, 2013], scheduling methods are proposed for noncyclic scheduling problem for single-arm cluster tools. With small batch, lot switching occurs frequently. Thus, studies are conducted and techniques are developed for scheduling lot switching processes for both single and dual-arm cluster tools [Lee et al., 2012 and 2013; and Kim et al., 2013b and 2013c].

However, all the above studies about scheduling a transient process in a cluster tool are not applicable for a single-arm cluster tool with wafer residency time constraints, which are not considered in [Lee et al., 2012 and 2013; Kim et al., 2013a, 2013b, and 2013c; and Wikborg and Lee, 2013]. Such constraints can make an optimal schedule for a transient process without residency time constraints considered infeasible. With wafer residency time constraints, Kim et al. [2012] propose scheduling methods to minimize the transient period for the start-up and close-down processes for dual-arm cluster tools. Since different scheduling strategies are required to schedule single-arm cluster tools. Their research results cannot be used to find an optimal feasible transient process for residency time-constrained single-arm cluster tools.

There is a need in the art to derive a solution to this optimal feasible transient process and to develop a method for scheduling a single-arm cluster tool based on the derived optimal solution.

SUMMARY OF THE INVENTION

The present invention provides a method for scheduling a cluster tool. The cluster tool comprises a single-arm robot for wafer handling, a LL for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint.

The method includes scheduling a start-up process for the cluster tool. The start-up process is developed based on Scheduling Algorithm 1 and the LPM model detailed below.

Preferably, the method further includes scheduling a steady-state process according to results obtained in the start-up process.

Other aspects of the present invention are disclosed as illustrated by the embodiments hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A depicts an example of a cluster tool having a single-arm robot.

FIG. 1B depicts an example of a cluster tool having a dual-arm robot.

FIG. 2 depicts a Petri net (PN) for the steady state of a single-arm cluster tool according to [Wu et al., 2008].

FIG. 3 depicts a Petri net for the start-up process of a single-arm cluster tool.

FIG. 4 shows the simulation results for Example 1.

FIG. 5 shows the simulation results for Example 2.

DETAILED DESCRIPTION

A PN model is developed for the start-up process of a single-arm cluster tool in Section A. Section B recalls the schedulability conditions and scheduling analysis for single-arm cluster tools [Wu et al., 2008]. Then, a scheduling algorithm and a linear programming model are developed for the start-up transient process scheduling in Section C.

Hereinafter, the notation N_(n), n being a positive integer, denotes a set containing positive integers from 1 to n, i.e. N_(n)={1, 2, . . . , n}.

A. PETRI NET MODELING AND CONTROL A.1. Finite Capacity Petri Nets

As an effective tool, PNs are widely used in modeling, analysis, and control of discrete-event systems, process industry, and robotic control systems [Zhou and DiCesare, 1991; Zhou et al., 1992 and 1995; Tang et al., 1995; Simon et al., 1998; Caloini et al., 1998; Zhou and Jeng, 1998; Wu and Zhou, 2001 and 2004; Liao et al., 2004; Ferrarini and Piroddi, 2008; Jung and Lee, 2012; Wu et al., 2008b; and Liu et al., 2013]. Following Zhou and Venkatesh [1998], the present work adopts a finite capacity PN to model a single-arm cluster tool. It is defined as PN=(P, T, I, O, M, K), where P={p₁, p₂, . . . , p_(m)} is a finite set of places; T={t₁, t₂, . . . , t_(n)} is a finite set of transitions with P∩T≠Ø and P∩T=Ø; I:P×T→N={0, 1, 2, . . . } is an input function; O: P×T→N is an output function; M: P→N is a marking representing the number of tokens in places with M₀ being the initial marking; and K: P→N\{0} is a capacity function where K(p) represents the largest number of tokens that p can hold at a time.

The preset of transition t is the set of all input places to t, i.e. t={p: p∈P and I(p, t)>0}. Its postset is the set of all output places from t, i.e., t={p: p∈P and O(p, t)>0}. Similarly, p's preset p={t∈T: O(p, t)>0} and postset p={t∈T: I(p, t)>0}. The transition enabling and firing rules can be found in [Wu and Zhou, 2009].

A.2. PN Model for Cluster Tools

In the present work, it is assumed that there are n≧2 steps in a cluster tool and only one PM serves for each step. Let (PM₁, PM₂, . . . , PM_(n)) denote the wafer flow pattern, where PM_(i), i ∈ N_(n), represents a process model being used to process wafers at Step i. Thus, a wafer needs to be processed at PM₁−PM_(n) sequentially before it is completed. Wu et al. [2008a] developed a PN model and conducted the steady periodical scheduling analysis for a single-arm cluster tool with wafer residency time constraints. We briefly introduce their PN model next.

In such a PN model, Step i is modeled by timed place p_(i) with K(p_(i))=1, i ∈ N_(n). The LLs are treated just as a processing step called Step 0. Since the LLs can hold all the wafers in a tool, they are modeled by p₀ with K(p₀)=∝. The robot is modeled by place r with K(r)=1, meaning that it has only one arm and can hold one wafer at a time. When M(r)=1, it represents that the robot arm is available. When M(p_(i))=1, i ∈ N_(n), a wafer is being processed in the PM for Step i. When the robot arrives at Step i for unloading a wafer, the wafer may be under way. Then, it has to wait there for some time. Timed place q_(i), i ∈ N_(n), is added to model the robot's waiting at Step i before unloading a wafer there and M(q_(i))=1 means that the robot is waiting at Step i. Note that the explicit representation of a robot wait as a place is critically important to deal with residency time constraints. Non-timed place z_(ij) is used to model the state at which it is ready to load a wafer to Step i or the wafer unloading from Step i ends. Transitions are used to model the robot tasks. Timed t_(i1), i ∈ N_(n), models loading a wafer into Step i, and t₀₁ models loading a completed wafer into a LL. Timed t_(i2), i ∈ N_(n), models unloading a wafer from Step i, and t₀₂ models unloading a raw wafer from a LL. Timed transition y_(i), i ∈ N_(n−2)∪{0}, represents the robot's moving from Steps i+2 to i without carrying a wafer; while transitions y_(n−1) and y_(n) represent the robot's moving from a LL to Step n-1 and Steps 1 to n, respectively. Timed transition x_(i), i ∈ N_(n−1)∪{0}, models the robot's moving from Steps i to i+1 with a wafer held, and x_(n) models the robot's moving from Steps n to 0. Pictorially, p_(i)'s and q_(i)'s are denoted by

, z_(ij)'s by ◯, and r's by

. Then, the PN model for a single-arm cluster tool is shown in FIG. 2.

At the steady state, every process module has one wafer being processed, i.e., Σ_(i=1) ^(n) K(p_(i)) wafers are being processed. For the PN model in FIG. 2, consider marking M with M(p_(i))=1, i ∈ N_(n), and M(r)=1. At this marking, y₀ is enabled and firing y₀ leads the PN to a dead marking, or the PN is deadlock-prone. Thus, according to [Wu et al., 2008], a control policy is proposed to make it deadlock-free.

Control Policy 1 (CP1): At any M of the PN model in FIG. 2, y_(i), i ∈ N_(n−1)∪{0}, is said to be control-enabled if M(p_(i+1))=0; and y_(n) is said to be control-enabled if M(p_(i))=1, i ∈ N_(n).

Before a cluster tool reaches its steady state, it must experience a start-up process. For a single-arm cluster tool, because the processing time is much longer than the robot task time, a backward strategy is found to be optimal [Lee et al., 2004; and Lopez and Wood, 2003]. Thus, a backward strategy is also used to operate the single-arm cluster tool for the start-up process. At the initial state, there is no wafers being processed in the tool, or the tool is empty. Let M_(s0) denote the initial state. When the tool starts to work, the robot unloads a wafer from the LLs, moves to Step 1, and loads this wafer into Step 1. Let M_(s1) denote the state of the system when the robot finishes the robot task of loading the wafer into Step 1. Then, the robot should wait there till this wafer is completed. After the wafer is processed, the robot unloads this wafer from Step 1 as soon as possible, moves to Step 2, loads this wafer into Step 2, returns to the LLs and unloads a raw wafer from the LLs, moves to Step 1, and loads the raw wafer into Step 1. At this time, Step 1 and Step 2 both have one wafer being processed. Thus, let M₅₂ denote the state of the system at this time. In the following operations of the system, the tool would reach a state that the Step i, i ∈N_(d) and d<n, has one wafer being processed and Step i, d<i≦n, is empty. To model this state, a PN model is developed shown in FIG. 3.

The places in the PN model in FIG. 3 have the same meanings as the ones in FIG. 2. Transitions t_(i1), ∈ N_(d+1), d<n and d≧2, t_(i2), i ∈ N_(d)∪{0}, d<n and d≧2, x_(i), i ∈ N_(d)∪{0}, d<n and d≧2, and y_(i), i ∈ N_(d−1)∪{0}, d<n and d≧2, in the PN model in FIG. 3 also have the same meanings as the ones in FIG. 2. Transition y_(d) represents the robot's moving from Steps 1 to d. Because Step i, i ∈ N_(d), d<n and d≧2, has one wafer being processed and Step i, d<i≦n, is empty, we have M(p_(i))=K(p_(i)), i ∈ N_(d), d<n and d≧2, and M(r)=1. At the marking shown in FIG. 3, y₀ is enabled and can fire. It can be seen that firing y_(o) leads the PN to a dead marking, or the PN is deadlock-prone. Then, a control policy is introduced to make it deadlock-free.

Control Policy 2 (CP2): For the PN model in FIG. 3, y_(i), i ∈ N_(d)∪{0}, d<n and d≧2, is said to be control-enabled if M(p_(i+1))=0.

With CP2, the start-up process could be described by running the PN model in FIG. 3 shown as follows. At the state M_(s2), Steps 1 and 2 both have one wafer being processed and the robot stays at Step 1. To describe the state M_(s2), we can set d=2 in the PN model and M_(s2)(p_(i))=K(p_(i)), i ∈ N₂, and M_(s2)(r)=1 holds. According to CP2, the following transitions firing sequence is: firing y₂ (moving to Step 2)→firing t₂₂ (unloading a wafer from Step 2)→firing x₂ (moving from Steps 2 to 3)→firing t₃₁ (loading the wafer into Step 3)→firing y₁ (moving from Steps 3 to 1)→firing t₁₂ (unloading a wafer from Step 1)→firing x₁ (moving from Steps 1 to 2)→firing t₂₁ (loading the wafer into Step 2)→firing y₀ (moving from Steps 2 to 0)→firing t₀₂ (unloading a wafer from Step 0)→firing x₀ (moving from Steps 0 to 1)→firing t₁₁ (loading the wafer into Step 1). At this time, the system reaches state M_(s3) such that M_(s3)(p_(i))=K(p_(i)), i ∈ N₃, and M_(s3)(r)=1 hold. Then, we can set d=3 in the PN model. According to CP2, the PN model can evolve to state M_(s4) with M_(s4)(p_(i))=K(p_(i)), i ∈ N₄, and M_(s4)(r)=1. Similarly, with the PN model in FIG. 3 and CP2, the PN model can evolve to state M_(sn) with M_(sn)(p_(i))=K(p_(i)), i ∈ N_(n), and M_(sn)(r)=1. At this time, the cluster tool is full of wafers and it reaches the steady state.

A.3. Activity Time Modeling

In the PN models in FIGS. 2 and 3, to describe the temporal aspect of a cluster tool, both transitions and places are associated with time. We use μ to denote the time for the robot task of moving with or without carrying a wafer. Time a_(i) is used to denote the time taken for completing a wafer at a step. It is assumed that the time taken for the robot's unloading a wafer from a step and loading a wafer into a step/LL is same and denoted by α. Transition t₀₂ models the robot's unloading a wafer from the LLs and aligning a wafer. Therefore, the time associated with t₀₂ is α₀ that is different from α. The robot's waiting time (denoted by ω_(i)) in q_(i) is determined by a schedule and can be a real number in [∝), or ω_(i) ∈[0, ∝). The detailed explanation of temporal features is summarized in Table 1.

TABLE 1 The Time Durations Associated with Transitions and Places Allowed Transition time or place Actions duration t_(i1) Robot loads a wafer into Step i, i∈N_(n)∪{0} α t_(i2) Robot unloads a wafer from Step i, i∈N_(n) t₀₂ Robot unloads a wafer from a LL and aligns it α₀ y_(i) Robot moves from a step to another without μ carrying a wafer x_(i) Robot moves from a step to another with a wafer carried p_(i) A wafer being processed in p_(i), i∈N_(n) a_(i) q_(i) robot waits before unloading a wafer from Step i, ω_(i) i∈N_(n)∪{0} z_(ij) No activity is associated

With wafer residency time constraints, the deadlock-freeness does not mean that the PNs shown in FIGS. 2 and 3 are live, because a token in p_(i) cannot stay there beyond a given time interval. Let τ_(i) be the sojourn time of a token in p_(i) and δ_(i) the longest time for which a wafer can stay in p_(i) after it is processed. Then, the liveness of the PN model is defined as follows.

Definition 1: The PN models in FIGS. 2 and 3 for single-arm cluster tools with residency time constraints are live, if 1) at any marking with a token in p₁, ∀i ∈ N_(n), and when t_(i2) fires a_(i)≦τ_(i)≦a_(i)+δ_(i) holds; 2) at any marking with a token in p_(i), i ∈N_(d) and d<n, and when t_(i2) fires a_(i)≦τ_(i)≦a_(i)+δ_(i) holds, respectively.

B. SCHEDULABILITY CONDITIONS

Before scheduling the start-up process, we recall the necessary and sufficient schedulability conditions of a single-arm cluster tool with wafer residency time constraints under the steady state derived in [Wu et al., 2008].

B.1. Timeliness Analysis for the Steady State

It follows from [Wu et al., 2008] that, to complete the processing of a wafer at Step i, i ∈ N_(n−1), it takes τ_(i)+4α+3μ+ω_(i−1) time units, where τ_(i) should be within [a_(i), a_(i)+δ_(i)]. With only one PM at Step i, i ∈ N_(n), we have that the lower permissive cycle time at Step i is

θ_(iL) =a _(i)+4α+3μ+ω_(i−1) , i ∈ N _(n)\{1}  (1)

The upper permissive cycle time at Step i is

θ_(1L) =a _(i)+4α+3μ+ω_(i−1)+δ_(i) , i ∈ N _(n)\{1}.   (2)

For Step 1, the lower one is

θ_(1L) =a ₁+3α+α₀+3μ+ω₀.   (3)

Its upper one is

θ_(1U) =a ₁+3α+₀+3μ+ω₀+δ₁.   (4)

It follows from (1)-(4) that the robot waiting time ω_(i), E ∈ N_(n−1)∪{0}, affects the permissive wafer sojourn time. Thus, by carefully regulating them, one can change the permissive range among the steps. By removing them from the above expressions, we obtain the lower and upper workloads with no robot waiting for each step as follows:

_(iL) =a _(i)+4α+3μ, i ∈ N _(n)\{1},   (5)

_(iU) =a _(i)+4α+3μ+δ_(i) , i ∈ N _(n)\{1},   (6)

_(1L) =a _(i)+3α+α₀+3μ  (7)

and

_(1U) =a ₁+3α+α₀+3μ+δ₁,   (8)

where

_(jL) and

_(jU) are the lower and the upper workloads, respectively, for Step j, j ∈ N_(n).

To schedule a single-arm cluster tool with residency time constraints, one has to ensure a_(i)≦τ_(i)≦a_(i)+δ_(i). Hence, we need to know how τ_(i) is calculated. According to [Wu et al., 2008], we have that

$\begin{matrix} {{{\tau_{i} = {{{2\left( {n + 1} \right)\mu} + {\left( {{2\; n} + 1} \right)\alpha} + \alpha_{0} + {\sum\limits_{d = 0}^{n}\; \omega_{d}} - \left( {{4\alpha} + {3\mu} + \omega_{i - 1}} \right)} = {\psi - \left( {{4\alpha} + {3\mu} + \omega_{i - 1}} \right)}}},\mspace{79mu} {i \in {N_{n}\backslash \left\{ 1 \right\}}}}\; \mspace{20mu} {and}} & (9) \\ {\tau_{1} = {{{2\left( {n + 1} \right)\mu} + {\left( {{2\; n} + 1} \right)\alpha} + \alpha_{0} + {\sum\limits_{d = 0}^{n}\; \omega_{d}} - \left( {{3\alpha} + \alpha_{0} + {3\mu} + \omega_{0}} \right)} = {\psi - {\left( {{3\alpha} + \alpha_{0} + {3\mu} + \omega_{0}} \right).}}}} & (10) \end{matrix}$

The robot cycle time is given by

ψ=2(n+1 )μ+(2n+1)α+α₀+Σ_(d=0) ^(n)ω_(d)=ψ₁+ψ₂   (11)

where ψ₁=2(n+1)μ+(2n+1)α+α₀ is a known constant and ψ₂=Σ_(d=0) ^(n)ω_(d) is to be decided by a schedule. It should be pointed out that ψ is independent of the ω_(i)'s. Let θ₁=τ₁+3α+α₀+3μ+ω₀ and θ_(i)=τ_(i)+4α+3μ+ω_(i−1), i ∈ N_(n)−{1}, denote the cycle time for Step i, i ∈ N_(n). Then, it can be seen that, by making ω_(i−1)>0, the cycle time of Step i is increased without increasing the wafer sojourn time. Thus, it is possible to adjust the robot waiting time to balance the wafer sojourn time among the steps such that a feasible schedule can be obtained. For a periodic schedule in a steady state, we have

θ=θ₁=θ₂= . . . =θ_(n)=ψ.   (12)

In (11), μ, α, and α₀ are all deterministic, only ω_(d), d ∈ N_(n)∪{0}, are changeable, i.e., ψ₁ is deterministic and ψ₂ can be regulated. Thus, based on the PN model shown in FIG. 2, one can schedule a single-arm cluster tool with residency time constraints by appropriately regulating ω_(d), d ∈ N_(n)∪{0}, such that (12) holds and at the same time the wafer residency time constraints are fully satisfied.

B.2. Schedulability Conditions for the Steady State Scheduling

To find a feasible cyclic schedule, the key is to know under what conditions there exists θ such that the system is schedulable. Notice that, in (5)-(8),

_(iL) and

_(iU) denote the lower and upper bounds of θ_(i), respectively. Let

_(max)=max{

_(iL), i ∈N_(n)}. Then, Wu et al. [2008] establish the following schedulability conditions.

Theorem 1:

_(max)≦

_(iU) and ψ₁≦

_(iU), i ∈ N_(n), a single-arm cluster tool with residency time constraints is schedulable.

For this case, when

_(max)≦

_(iU) and ψ₁≦

_(max), i ∈ N_(n), the tool is process-bound. When

_(iL)≦ψ₁≦

_(iU), i ∈ N_(n), a tool is transport-bound. With

_(max)≦

_(iU), i ∈ N_(n), the difference of the workloads among the steps is not too large. Thus, by properly setting ω_(i)'s, the workloads among the steps can be balanced such that there is a feasible cyclic schedule. It follows from [Wu et al., 2008] that, in this case, one can simply set ω_(i)=0, i ∈ N_(n−1)∪{0}, and ω_(n)=max{

_(max)−ψ₁, 0} such that ψ=max{

_(max), ψ₁} holds. In this way, a feasible schedule is found. Further, it is optimal in terms of cycle time.

By Theorem 1, to make the tool schedulable requires that the workloads among the steps are not too large, i.e. [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]≠Ø. However, sometimes we have [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]≠Ø. In this case, let E={i|i ∈ N_(n),

_(iU)<

_(max)} and F=N_(n)\E. It follows from [Wu et al., 2008] that the time for completing a wafer at Step i can be increased by setting ω_(i−1)>0 without changing sojourn time τ_(i). Hence, a cluster tool may be made schedulable even if the workloads among the steps are not well balanced. To do so, we balance the workloads among the steps by setting ω_(i−1)'s as follows:

$\begin{matrix} {\omega_{i - 1} = \left\{ {\begin{matrix} {0,} & {i \in F} \\ {{\vartheta_{\max} - \left( {a_{1} + \delta_{1} + {3\alpha} + \alpha_{0} + {3\mu}} \right)},} & {1 \in E} \\ {{\vartheta_{\max} - \left( {a_{i} + \delta_{i} + {4\alpha} + {3\mu}} \right)},} & {1 \in {E\bigcap\left\{ {2,3,4,\ldots \mspace{14mu},n} \right\}}} \end{matrix}.} \right.} & (13) \end{matrix}$

Theorem 2: If [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]≠Ø,

_(IU)<

_(max), i ∈ E≠Ø,

_(iU)≧

_(max), i ∈ F, and Σ_(i∈F.)ω_(i−1)+ψ₁≦

_(max), a single-arm cluster tool with residency time constraints is schedulable with ω_(i−1), i ∈ N_(n), being set by (13).

In this case, with the robot waiting time ω_(i−1), i ∈N_(n), being set by (13), without changing τ_(i), θ_(i) for completing a wafer at Step i can be increased such that the workload among the steps can be properly balanced. Notice that, by (13), the robot waiting time ω_(i−1), i ∈ N_(n), is set, and then let ω_(n)=

_(max)−(ψ₁+Σ_(i∈E)ω_(i−1)) such that ψ=θ_(max) hold. Thus, a feasible schedule is obtained and the cycle time is optimal. According to [Wu et al., 2008], the conditions given by Theorems 1 and 2 are the necessary and sufficient schedulability conditions for a single-arm cluster tool with residency time constraints. In the next section, we conduct the start-up process scheduling analysis for the system.

C. START-UP PROCESS SCHEDULING C.1. Temporal Properties in Start-up Process

At the initial state denoted by M_(s0), the cluster tool is idle. When the tool starts to work, the robot unloads a wafer from the LLs, moves to Step 1, and loads this wafer into Step 1. At this time, M_(s1) is reached. From states M_(s0) to M_(s1), it takes (α₀+μ+α) time units. Then, the robot should wait there for a₁ time units before the wafer in Step 1 is completed. Then, (α₀+3α+3μ) time units would be taken for performing the following robot task sequence: unloads this wafer from Step 1 as soon as possible, moves to Step 2, loads this wafer into the Step 2, returns to the LLs and unloads a raw wafer from the LLs, moves to Step 1, and loads the raw wafer into Step 1. At this time, both Steps 1 and 2 have one wafer being processed and M_(s2) is reached. From M_(s1) to M_(s25) it takes (α₁+α₀+3α+3μ) time units.

Observing the PN model shown in FIG. 3, we have that M_(sd)(p_(i))=K(p_(i)), i ∈ N_(d) and 2≦d≦n−2, and M_(sd)(r)=1. With CP2, the PN model can evolve to state M_(s(d+1)) with M_(s(d+1))(p_(i))=K(p_(i)), i ∈N_(d+1), and M_(s(d+1))(r)=1, and then to M_(s(d+2)) with M_(s(d+2))(p_(i))=K(p_(i)), i ∈ N_(d+2), and M_(s(d+2))(r)=1. During the evolution from M_(s(d+1)) to M_(s(d+2)), the robot should sequentially go to Steps (d+1), d, . . . , and 1 for unloading the processed wafers. Thus, with wafer residency time constraints, it is necessary to know how much time is needed to complete the processing of a wafer at Step i, i ∈N_(d+1). Notice that the wafers unloaded from Step i, i ∈N_(d+1), during the period from M_(s(d+1)) to M_(s(d+2)) are all the ones loaded into Step i, i ∈ N_(d+1), during the period from M_(sd) to M_(s(d+1)). Thus, from the PN model in FIG. 3, CP2, and the period from M_(sd) to M_(s(d+2)), to complete the processing of a wafer in Step i, i ∈N_(d+1), the following transition firing (activities) sequence must be executed: Firing t_(i2) (time α)→x_(i) (time μ)→t_((i−1)1) (time α)→y_(i−1) (time μ)→robot waiting in q_(i−1) (time ω_(i−1))→t_((i−1)2) (time α)→x_(i−1) (time μ)→t_(i1) (time α)→processing a wafer at Step i (time τ_(i))→t_(i2) (time α) again. In this way, a cycle is completed and it takes (τ_(i)+4α+3μ+ω_(i−1)) time units to complete a wafer. In fact, for a wafer unloaded from Step i, i ∈ N_(d), during the period from M_(sd) to M_(s(d+1)), 2≦d<n, it also takes (τ_(i)+4α+3μ+ω_(i−1)) time units to complete this wafer. Notice that τ_(i) should be within [a_(i), a_(i)+δ_(i)]. When τ_(i)=a_(i), we have the lower permissive time to complete a wafer at Step i, i ∈N_(d)\{1}, which equals to the one given by (1). When τ_(i)=a_(i)+δ_(i), we have the upper permissive time to complete a wafer at Step i, which equals to the one given by (2). For Step 1, the lower and upper permissive time to complete a wafer equals to the ones in (3) and (4), respectively. Then, if the robot waiting time is removed from (1)-(4), the lower and upper workloads without robot waiting for each step can be obtained via (5)-(8), respectively.

EQNS. (5)-(8) present the workload balance information that affects the existence of a feasible schedule. It follows from (2) and (6) that θ_(iU)>

_(iU) if ω_(i−)>0. It implies that, by making ω_(i−1)>0, the cycle time of Step i is increased without increasing the wafer sojourn time. Thus, it is possible to adjust the robot waiting time such that the permissive wafer sojourn time ranges among the steps are balanced to some extent to guarantee the feasibility. To do so, we need to know how τ_(i) should be calculated. The wafer sojourn time at p_(i) depends on the robot tasks and the workloads of the steps. From the PN model shown in FIG. 3 and CP2, during the evolutions from M_(s(d−1)) to M_(sd), a wafer (W₁) is loaded by the robot into Step i: Firing y_(d−1)→waiting in q_(d−1)→firing t_((d−1)2)→firing x_(d−1)→firing t_(d1)→firing y_(d−2)→waiting in q_(d−2)→firing t_((d−2)2)→firing x_(d−2)→firing t_((d−1)1)→ . . . →firing y_(i)→waiting in q_(i)→firing t_(i2) with unloading a wafer from Step i→firing x_(i)→firing t_((i+1))→firing y_(i−1)→waiting in q_(i−1)→firing t_((i−1)2)→firing t_(i−1)→firing t_(i1) to load wafer W₁ into Step i→ . . . →y₁→waiting in q₁→firing t₁₂→x₁→firing t₂₁→firing y₀→waiting in q₀→firing t₀₂→firing x₀→firing t₁₁. Then, from the PN model shown in FIG. 3 and CP2, during the evolutions from M_(sd) to M_(s(d+1)), wafer W₂ is unloaded by the robot from Step i: Firing y_(d)→waiting in q_(d)→firing t_(d2)→firing x_(d)→firing t_((d+1)1)→firing y_((d−1))→waiting in q_((d−1))→firing t_((d−1)2)→firing x_(d−1)→firing t_(d1)→ . . . →firing y_(i)→waiting in q_(i)→firing t_(i2) to unload wafer W₁ from Step i→firing x_(i)→firing t_((i+1)1)→firing y_(i−1)→waiting in q_(i−1)→firing t_((i−1)2)→firing x_(i−1)→firing t_(i1)→ . . . →y₁→waiting in q₁→firing t₁₂→x₁→firing t₂₁→firing y₀→waiting in q₀→firing t₀₂→firing x₀→firing t₁₁. Thus, from the above PN evolutions we have that, during the evolution from M_(sd) to M_(s(d+1)), 2≦d<n, the wafer sojourn time in p₁, i ∈N_(d), is given by

$\begin{matrix} {{\tau_{1} = {{2\left( {d + 1} \right)\mu} + {\left( {{2\; d} + 1} \right)\alpha} + \alpha_{0} + {\sum\limits_{j = 0}^{d}\; \omega_{j}} - \left( {{3\alpha} + \alpha_{0} + {3\mu} + \omega_{0}} \right)}}\mspace{20mu} {and}} & (14) \\ {{\tau_{i} = {{2\left( {d + 1} \right)\mu} + {\left( {{2\; d} + 1} \right)\alpha} + \alpha_{0} + {\sum\limits_{j = 0}^{d}\; \omega_{j}} - \left( {{4\alpha} + {3\mu} + \omega_{i - 1}} \right)}},\mspace{20mu} {1 < i \leq {d.}}} & (15) \end{matrix}$

Let ψ_(sd(d+1)) and ψ_(sd(d+1)1) denote the robot task time for transferring the tool from states M_(sd) to M_(s(d+1)) with and without robot waiting time considered, respectively. Thus, we have

$\begin{matrix} {{\psi_{{sd}{({d + 1})}} = {{2\left( {d + 1} \right)\mu} + {\left( {{2\; d} + 1} \right)\alpha} + \alpha_{0} + {\sum\limits_{j = 0}^{d}\; \omega_{j}}}}{and}} & (16) \\ {\psi_{{{sd}{({d + 1})}}1} = {{2\left( {d + 1} \right)\mu} + {\left( {{2\; d} + 1} \right)\alpha} + {\alpha_{0}.}}} & (17) \end{matrix}$

It follows from (14)-(17) that to schedule the transient process of a residency-time constrained single-arm cluster tool is to appropriately regulate ω_(j), j ∈ N_(d)∪{0}, such that the wafer residency time constraints at each step are all satisfied.

C.2. Scheduling for Start-up Process

Feasibility is an essential requirement for scheduling a transient process of a cluster tool. As we have mentioned that, at initial state M_(s0), the cluster tool is idle. When the tool reaches states M_(s1) to M_(sn), Steps 1 to n have one wafer being processed, respectively. For the start-up process, the robot tasks are determined. Thus, we just need to determine the robot waiting time during the period from M_(s0) to M_(sn) to find a feasible schedule for the start-up process. Then, we have the following schedulability proposition.

Proposition 1: A start-up process of a single-arm cluster tool with wafer residency time constraints is schedulable if there exists the robot waiting time setting during the period from M_(s0) to M_(sn) such that the wafer residency time constraint at each step is satisfied.

With Proposition 1, we know that it is necessary to propose a method to regulate the robot waiting time during the period from M_(s0) to M_(sn) such that the cluster tool can enter the desired steady state from the initial state without violating the wafer residency time constraints.

In a cluster tool, it is reasonable to assume that there are more than one processing step. For the tool with two processing steps, the start-up process from M_(s0) to M_(s2) could be described by a robot task sequence σ₁: Unloading a raw wafer (W₂) from the LLs (time α₀)→moving to Step 1 (time μ)→loading wafer W₂ into Step 1 (time α)→waiting at Step 1 for ω₁=a₁ time units→unloading wafer W₂ from Step 1 (time α)→moving to Step 2 (time μ)→loading wafer W₂ into Step 2 (time α)→moving to the LLs (time μ)→waiting at the LLs for ω₀ time units→unloading a raw wafer (W₃) from the LLs (time α₀)→moving to Step 1(time μ)→loading wafer W₃ into Step 1 (time α). At this time, the system reaches state M_(s2). In σ₁, only robot waiting time ω₀ is unknown. Let |σ₁| denote the time needed to perform sequence σ₁. Thus, |σ₁|=2α₀+4α+4μ+a₁+ω₀. Therefore, for the single-arm cluster tool, to obtain a feasible start-up schedule is to determine the robot waiting time ω₀. For the single-arm cluster tool with n>2 processing steps, during the process from M_(s0) to M_(s2), the robot task sequence is also σ₁. Then, the system keeps working according to the PN model in FIG. 3 and CP2 till it reaches state M_(sn). For this case, the robot waiting time is also unknown during the process from M_(s0) to M_(sn). Let

_(max)=max{

_(iL), i ∈ N_(d)}. To solve the scheduling problem, the following algorithm is developed.

Scheduling Algorithm 1: If

_(max)≦

_(iU) and ψ₁≦

_(iU), i ∈ N_(n), the robot waiting time is set as follows.

Situation 1: A single-arm cluster tool has two steps:

-   -   1) During the process from M_(s0) to M_(s2), the tool operates         according to the robot task sequence σ_(i) and set ω₀=0 and ω₁=1         ₁.

Situation 2: A single-arm cluster tool has n steps, n>2:

-   -   1) During the process from M_(s0) to M_(s2), the performance of         the tool is same as the one regulated by 1) in Situation 1;     -   2) During the process from M_(sd) to M_(s(d+1)), 2≦d<n, the tool         operates according to the PN model in FIG. 3 and CP2 and set         ω_(i)=0, i ∈ N_(d−1)∪{0}, and ω_(d)=max {         _(dmax)−ψ_(sd(d+1)1), 0}.

In this case, there are two situations. For Situation 1, there are two steps in a single-arm cluster tool. Then, during the start-up process from M_(s0) to M_(s2), the tool operates according to the robot task sequence σ₁, and the robot waiting time ω₀ and ω₁ in σ₁ can be set as ω₀=0 and ω₁=a₁. With M_(s2) being reached, the system reaches its steady state. For Situation 2, a single-arm cluster tool has n steps, n>2. From M_(s0) to M_(s2), the performance of the tool is same as the one regulated by 1) in Situation 1. Then, during the process from M_(sd) to M_(s(d+1)), 2≦d<n, the tool operates according to the PN model in FIG. 3 and CP2. At the same time, the robot waiting time is set as ω_(i)=0, i ∈ N_(d−1)∪{0}, and ω_(d)=max{

_(dmax)−ψ_(sd(d+1)1), 0} such that ψ_(sd(d+1))=max{

_(dmax), ψ_(sd(d+1)1)}. When the tool reaches state M_(sn), the system reaches its steady state. Then, by the PN model in FIG. 2 and CP1, the system operates with the backward strategy. By Scheduling Algorithm 1, a schedule could be found for the start-up process. Then, the next question is if this obtained schedule is feasible. The following theorem answers it.

Theorem 3: For a single-arm cluster tool with wafer residency time constraints, if

_(max)≦

_(iU), ψ₁≦

_(iU), i ∈ N_(n), a schedule obtained by Scheduling Algorithm 1 is feasible.

Proof Consider Situation 1. For the start-up process from M_(s0) to M_(s2), the robot performs the robot tasks σ₁. It is easy to find that wafer W₂ can be unloaded from Step 1 without violating the residency time constraints. Then, W₃ is delivered to Step 2. When M_(s2) is reached, the system enters its desired steady state. Consider Situation 2. By 1) for Situation 2 of Algorithm 1, similarly, the robot can perform the robot tasks c_(h) such that the cluster tool can reach M_(s2) from M_(so) without violating the wafer residency time constraints. Then, by 2) for Situation 2 of Algorithm 1, the tool operates according to the PN model in FIG. 3 and CP2 during the process from M_(sd) to M_(s(d+1)), 2≦d<n. At the same time, ω_(i)=0, i ∈ N_(d−1)∪{0}, and ω_(d)=max{

_(dmax)ψ_(sd(d+1))₁, 0}. It follows from (14) and (15) that τ₁=2(d+1)μ+(2d+1)α+α₀+τ_(j=0) ^(d)ω_(j)−(3α+α₀+3μ+ω₀)=2(d+1)μ+(2d+1)α+α₀+max{

_(max)−ψ_(sd(d+1)1), 0}−(3α+α₀+3μ) and τ_(i)=2(d+1)μ+(2d+1)α+α₀+Σ_(j=0) ^(d)ω_(j)−(4α+3μ+ω_(i−1))=2(d+1)μ+(2d+1)α+α₀+max{

_(dmax)−ψ_(sd(d+1)1), 0}−(4α+α3μ), 1<i≦d, hold, respectively. Then, there are two cases. Case 1:

_(dmax)≦ψ_(sd(d+1)1). Then, we have τ₁=2(d+1)μ+(2d+1)α+α₀+

_(dmax)−ψ_(sd(d+1)1)−(3α+α₀+3μ) and τ_(i)=2(d+1)μ+(2d+1)α+α₀+

_(max)−ψ_(sd(d−1)1)−(4α++3μ), 1<i≦d, hold. Then, by (17) and

_(max)≦

_(iU), i ∈ N_(n), we have

_(1L)−(3α+α₀+3μ)≦τ₁=

_(dmax)−(3α+α₀+3μ)≦

_(1U)−(3α+α₀+3μ) and

_(iL)−(4α+3μ)≦τ_(i)=

_(dmax)−4α+3μ)≦

_(iU)−(4α+3μ), 1<i≦d. Thus, it follows from (5)-(8) that a₁=

_(1L)−(3α+α₀+3μ)≦τ₁≦

_(1U)−(3α+α₀+3μ)=a₁+δ₁ and a_(i)=

_(iL)−(4α+3μ)≦τ_(i)≦

_(1U)−(4α++3μ)=a_(i)+δ_(i), 1<i≦d, hold. Case 2:

_(dmax)<ψ_(sd(d+1)1). Then, from (17), we have τ₁=2(d+1)μ+(2d+1)α+α₀−(3α+α₀+3μ)=ψ_(sd(d+1)1)−(3α+α₀+3μ) and τ_(i)=2(d+1)μ+(2d+1)α+α₀−(4α+3μ)=ψ_(sd(d+1)1)−(4α+3μ), 1<i≦d, hold. By the assumption of ψ₁≦

_(iU), i ∈ N_(n), we have τ₁=ψ_(sd(d−1)1)−(3α+α₀+3μ)<ψ₁−(3α+α₀+3μ)≦≦

_(1U)−(3α+α₀+3μ) and τ_(i)=ψ_(sd(d+1)1)−(4α+3μ)<ψ₁−(4α+3μ)≦

_(iU)−(4α+3μ), 1<i≦d. Then, it follows from (5)-(8),

_(max)≦

_(iU), i ∈ N_(n), and

_(dmax)<ψ_(sd(d+1)1) that a₁=

_(1L)−(3α+α₀+3μ)≦

_(dmax)−(3α+α₀+3μ)<τ₁=ψ_(sd(d+1)1)−(3α+α₀+3μ)<

_(1U)−(3α+α₀+3μ)=a₁+δ₁ and a_(i)=

_(iL)−(4α+3μ)≦

_(dmax)−(4α+3μ)<τ_(i)=ψ_(sd(d+1)1)−(4α+3μ)<

_(iU)−(4α+3μ)=a_(i)+δ_(i), 1<i≦d. Therefore, during the start-up process from M_(s0) to M_(s2), the wafer residency time constraints at each step are all satisfied. Hence, the theorem holds.

In the case of Situation 1, by Algorithm 1, the robot performs σ_(i) such that the cluster tool can successfully go through the start-up process from M_(s0) to M_(s2) without violating any residency time constraints. Also, it takes |σ₁|=2α₀+4α+4μ+a₁ time units for the start-up process. In the case of Situation 2, by Algorithm 1, the schedule is same as the one before M_(s2) is reached. Then, by Algorithm 1, we need to dynamically adjust the robot waiting time at step d during the process from M_(sd) to M_(s(d+1)), 2≦d<n, such that ψ_(sd(d+1))=max{

_(dmax), ψ_(sd(d+1)1)}. Thus, it takes |σ₁|+Σ_(d=2) ^(n−1) max(

_(d max), ψ_(sd(d+1)1)) time units for the start-up process. For a single-arm cluster tool with n≧2 steps, when M_(sn) is reached, the system enters its desired steady state. In the following evolution, the system operates with the backward strategy. Based on Theorem 1, for the steady state scheduling, a feasible and optimal schedule is obtained by setting ω_(i)=0, i ∈N_(n−1)∪{0}, and ω_(n)=max{

_(max)−ψ₁, 0} such that ψ=max{

_(max), ψ₁} holds. Then, the following theorem proves its optimality.

Theorem 4: For a single-arm cluster tool with wafer residency time constraints, if

_(max)≦

_(iU), ψ₁≦

_(iU), i ∈ N_(n), a schedule obtained by Scheduling Algorithm 1 for the start-up process is optimal.

Proof Situation 1: For the start-up process from M_(s0) to M_(s2), the robot performs the robot tasks σ₁. If there be a schedule better than the one obtained by Algorithm 1, it must be that the robot waiting time ω₁ is shortened because of ω₀=0. However, if ω₁ is less than a₁, the wafer being processed at Step 1 cannot be processed. Therefore, for Situation 1, the obtained schedule by Algorithm 1 is optimal. For Situation 2, similar to Situation 1, the obtained schedule by Algorithm 1 for the process from M_(s0) to M_(s2) is optimal. It follows from Theorem 3 that during the process from M_(sd) to M_(s(d+1)), 2 s≦d<n, we have τ₁=2(d+1)μ+(2d+1)α+α₀+max{

_(dmax)−ψ_(sd(d+1)1), 0}−(3α+α₀+3μ) and τ_(i)=2(d+1)μ+(2d+1)α+α₀+max{

_(dmax)−ψ_(sd(d−1)1), 0}−(4α+3μ), 1<i≦d, hold, respectively. It is assumed that

_(dmax)=

_(kL), 1≦k≦d. Then, we have τ₁=2(d+1)μ+(2d+1)α+α₀+max{

_(1L)−ψ_(sd(d+1)1), 0}−(3α+α₀+3μ)

_(dmax)=

_(1L) and τ_(k)=2(d+1)μ+(2d+1)α+α₀+max{

_(kL)−ψ_(sd(d+1)1), 0}−(4α+3μ) if

_(dmax)=

_(kL), 1<k≦d. If

_(dmax)≧ψ_(sd(d+1)1), we have

_(dmax)=ψ_(sd(d+1)) by Algorithm 1. By (17) and (5)-(8), we have τ₁=

_(1L)−(3α+α₀+3μ)=a₁ if

_(dmax)=

_(1L) and τ_(k)=

_(kL)−(4α+3μ)=a_(k) if

_(dmax)=

_(kL), 1<k≦d, hold. This means that it takes ψ_(sd(d+1))=

_(kL) time units for the process from M_(sd) to M_(s(d+1)), 2≦d n. Thus, for the process from M_(sd) to M_(s(d+1)), 2 s≦d<n, the wafer sojourn time just equals to a_(k) at Step k. If there exists a schedule for the process from M_(sd) to M_(s(d+1)), 2≦d<n, better than the one obtained by Algorithm 1, there must exist Step k where the wafer sojourn time is less than a_(k). This means the wafer at Step k cannot be processed. If

_(dmax)<ψ_(sd(d+1)1), a better schedule for the process from M_(sd) to M_(s(d+1)), 2≦d<n, cannot be found because ψ_(sd(d+1)1) cannot be shortened. Therefore, the obtained schedule by Algorithm 1 for the process from M_(sd) to M_(s(d+1)), 2≦d<n, is also optimal. Hence, the theorem holds.

By Theorem 3, the workloads among the steps are properly balanced, i.e. [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]≠Ø. However, there is also another case with [

_(1L),

_(1U)] ∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø. Under the steady state, the cycle time is a constant. Then, a feasible schedule could be found by setting ω_(i−1)>0, i ∈ E, to reduce the wafer sojourn time i without changing the time for completing a wafer at Step i [Wu et al., 2008]. For the transient process, we have: 1) wafers are processed at Step i, i ∈ E, during the process from M_(sd) to M_(s(d+1)) and M_(s(d+1)) to M_(s(d+2)), 2≦d≦n−2, respectively; 2) the time taken for the process from M_(sd) to M_(s(d+1)) and M_(s(d+1)) to M_(s(d+2)) may be different. Thus, the key to find a feasible and optimal schedule for the process from M_(sd) to M_(s(d+2)) is to dynamically adjust the robot waiting time ω_(i−1). However, increasing and decreasing ω_(i−1) would decrease and increase the wafer sojourn time τ_(i), respectively. This makes it difficult to guarantee the feasibility and optimality at the same time. Thus, a linear programming model is developed to solve this problem. Let t_(ij) ^(d) and ω_(i) ^(d) denote the time when firing t_(ij) completes and the robot waiting at Step i before unloading a wafer during the process from M_(sd) to M_(s(d+1)), respectively. Then, we have a linear programming model.

Linear Programming Model (LPM): If [

_(1L),

_(1U)]∩[

_(2L),

_(2U)] . . . ∩[

_(nL),

_(nU)]=Ø and the system checked by Theorem 2 is schedulable under the steady state, then a schedule can be found by the following LPM:

$\begin{matrix} {\min \; {\sum\limits_{d = 1}^{n}\; {\sum\limits_{i = 0}^{d}\; \omega_{i}^{d}}}} & (18) \end{matrix}$

subject to

t ₁₁ ⁰=α₀+μ+α,   (19)

t ₁₂ ¹ =t ₁₁ ⁰+ω₁ ¹+α,   (20)

t _(i1) ^(d) =t _((i=1)2) ⁰+μ+α, 1≦i≦d+1 and 1≦d≦n−1,   (21)

t _(i2) ^(d) =t _((i+2)1) ^(d)+μ+ω_(i) ^(d)+α, 1≦i≦d−1 and 1≦d≦n−1,   (22)

t ₀₂ ^(d) =t _((d+2)1) ^(d)+μ+ω₀ ^(d)+α₀, 1≦d≦n−1,   (23)

t _(d2) ^(d) =t ₁₁ ^(d−1)+μ+ω_(d) ^(d)+α, 1≦d≦n−1,   (24)

t _(i1) ^(d) =t _((i+1)2) ^(d)+μ+α, 1≦i≦n and d ∈ {n,n+1},   (25)

t ₀₁ ^(d) =t _(n2) ^(d) +μα, ∈ {n, n+1},   (26)

t _(i2) ^(d) =t _((i+2)1) ^(d)+μ+ω_(i) ^(d)+α, 1≦i≦n−2 and d ∈ {n, n+1},   (27)

t _((n−1)2) ^(d) =t ₀₁ ^(d)+μ+ω_(n−1) ^(d) +α, d ∈ {n, n+1},   (28)

t _(n2) ^(d) =t ₁₁ ^(d−1)+μ+ω_(n) ^(d) +α, d ∈ {n, n+1},   (29)

t ₀₂ ^(d) =t ₂₁ ^(d)+μ+ω₀ ^(d)+α₀ , d ∈ {n, n+1},   (30)

ω_(i) ^(n)=ω_(l) ^(n+1), 0≦i≦n   (31)

$\begin{matrix} {{{\sum\limits_{d = 1}^{n}\; {\sum\limits_{i = 0}^{d}\; \omega_{i}^{d}}} \geq 0},} & (32) \end{matrix}$ and

a _(i) ≦t _(i2) ^(d) −α−t _(i1) ^(d−1) ≦a _(i)+δ_(i), 1≦i≦d and 1≦d≦n+1.   (33)

For a single-arm cluster tool with two processing steps, the robot task sequence for the start-up process from M_(s0) to M_(s2) is σ₁. Then, the system is operated with the backward strategy based on the PN model in FIG. 2. For a tool with n>2 processing steps, the robot task sequence for the start-up process from M_(s0) to M_(s2) is also σ₁. Then, the system will run according to the PN model in FIG. 3 till it reaches state M_(sn). In the following operations, the system is operated by the backward strategy based on the PN model in FIG. 2. Notice that the robot task sequence for the start-up process of the two cases is known in advance. However, the robot waiting time is unknown. If the waiting time during the process from M_(s0) to M_(sn) is determined, the schedule for the start-up process is determined. Objective (18) in LPM is to minimize the total robot waiting time. Constraints (19) and (21) give the time for completing the robot task of loading a wafer into a step. Constraints (20) and (22)-(24) represent the time for completing the robot task of unloading a wafer from a step. After M_(sn) is reached, the cluster tool enters its steady state and operates by the backward strategy based on the PN model in FIG. 2 and CP1. Then, when the first robot task cycle for the steady state is completed, state M_(s(n+1)) is reached. When the second robot task cycle for the steady state is completed, state M_(s(n+2)) is reached. Thus, for the first and second cycles for the steady state, Constraints (25) and (26) mean the time for completing the robot task of loading a wafer into a step, and Constraints (27)-(30) indicate the time for completing the robot task of unloading a wafer from a step. Constraint (31) makes sure that the robot waiting time is same for the different cycle under the steady state. Constraint (32) means that the robot waiting time is no less than zero. With wafer residency time constraints being considered, Constraint (33) is used to guarantee such constraints to be satisfied. In this way, a schedule for the start-up process could be obtained by this model.

For the case with [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, Theorem 2 gives schedulability conditions to check if the system is schedulable. Thus, it gives rise to a question that, for the case with [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, if the system checked by Theorem 2 is schedulable, can a feasible schedule be obtained by LPM? To answer it, a schedule can be obtained by setting the robot waiting time as: 1) For the tool with two processing steps, the robot waiting time can be set as ω₀ ^(d)=max{

_(max)−

_(1U), 0}, d ∈ {1, 2}, ω₁ ¹, =a₁, ω₁ ²=max{

_(max)−

_(2U), 0}, and ω₂ ²=

_(max)−ψ₁−(ω₀ ²+ω₁ ²); and 2) For the tool with more than n>2 processing steps, the robot waiting time can be set as ω_(i) ^(d)=max{

_(max)−

_((i+1)U), 0}, 0≦i≦n−1 and 2≦d≦n+1, ω₁ ¹=a₁, and ω_(d) ^(d)=

_(max)−ψ₁−Σ_(i=0) ^(d−1)ω_(i) ^(d), 2≦d≦n+1. It is easy to verify that this schedule is in the feasible region of LPM. Therefore, if the system is schedulable according to Theorem 2's conditions, a feasible and optimal schedule can be obtained by LPM. For the first and second cycles for the steady state, the robot waiting time ω_(i) ^(n) and ω_(i) ^(n+1), 0≦i≦n, can be determined by LPM. Then, in the following operations of the system under the steady state, the robot waiting time is also set as ω_(i)=ω_(i) ^(n), 0≦i≦n−1, and ω_(n)=

_(max)−Σ_(i=0) ^(n−1)ω_(i). Thus, another question is if the schedule for the steady state is feasible and optimal? The following theorem answers it.

Theorem 5: For a single-arm cluster tool with [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, with the PN in FIG. 2 and CP1, if a schedule for the steady state is given by setting ω_(i)=ω_(l) ^(n), 0≦i≦n−1, and ω_(n)=

_(max)−Σ_(i=0) ^(n−1)ω_(i), where ω_(i) ^(n), 0≦i≦n−1, is obtained by LPM, then such a schedule is feasible and optimal.

Proof By LPM, during the processes from M_(sn) to M_(s(n+1)) and M_(s(n+1)) to M_(s(n+2)), the robot waiting time is ω_(i)=ω_(i) ^(n), 0≦i≦n. Then, the cycle time for the processes from M_(sn) to M_(s(n+1)) and M_(s(n+1)) to M_(s(n+2)) should be ψ=

_(max). If there exists a schedule with the cycle time ψ<

_(max) and it is assumed that

_(max)=

_(kL), k≠1 holds, it follows from (9) that τ_(k)=[2(n+1)μ+(2n+1)α+α₀+Σ_(d=0) ^(n)ω_(d)]−(4α+3μ+ω_(k−1))=ψ−(4α+3μ+ω_(k−1))<

_(kL)−(4α+3μ+ω_(k−1))≦

_(kL)−(4α+3μ). Then, from (5), we have τ_(k)<

_(kL)−(4α+3μ)32 a_(k). This means that the wafer at Step k is not completed. Similarly, if there exists a schedule with the cycle time ψ<

_(max) and

_(max)=

_(1L) holds, we have τ₁<a₁. Therefore, the cycle time for the processes from M_(sn) to M_(s(n+1)) and M_(s(n+1)) to M_(s(n+2)) should be ψ=

_(max). This implies that Σ_(i=0) ^(n)ω₁=

_(max)−ψ₁. Thus, based on LPM and Theorem 2, this theorem holds and the cycle time of the system for the steady state is

_(max).

Up to now, for the case that the workloads among the steps can be properly balanced, i.e. [

_(1L),

_(1U)]∩[

_(2L),

_(2U)] ∩ . . . ∩ [

_(nL),

_(nU)]=Ø, a scheduling algorithm is proposed to find the optimal schedule for the start-up process such that the single-arm cluster tool can enter its steady state optimally. For the case that the differences of the workloads among the steps are too large such that [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, a linear programming model is developed to find a feasible and optimal schedule to transfer a single-arm cluster tool from the initial state to a steady one. Notice that Scheduling Algorithm 1 consists of several expressions and LPM is a linear programming model. Therefore, it is very computationally efficient to use the proposed methods to find a feasible and optimal schedule for the start-up process for single-arm cluster tools with wafer residency time constraints.

D. EXAMPLES Example 1

The flow pattern is (PM₁, PM₂, PM₃, PM₄, PM₅). It takes 5s for the robot to unload a wafer from a PM and to load a wafer to a PM/LL (α=5s), 10s to unload a wafer from the LLs and align it (α₀=10s), and 2s to move between PMs/LLs (μ=2s). It needs 90s, 100s, 100s, 105s, and 115s for a PM at Steps 1-5 to process a wafer (a₁=90s, a₂=100s, a₃=100s, a₄=105s, and a₅=115s), respectively. After being processed, a wafer at Steps 1-4 can stay there for 20s (δ₁=δ₂=δ₃=δ₄=δ₅=20s).

It follows from (5)-(8) that, we have

_(1L)=121s,

_(1U)=141s,

_(2L)=126s,

_(2U)=146s,

_(3L)=126s,

_(3U)=146s,

_(4L)=131s,

_(4U)=151s,

_(5L)=141s,

_(5U)=161s, and ψ₁=89s. By Theorem 1, the single-arm cluster tool is schedulable. For the steady state, an optimal and feasible schedule is obtained by setting ω₀=ω₂=ω₃=ω₄=0s and ω₅=52s. Then, the cycle time of the system under the steady state is 141s. For this example, the workloads among the steps are properly balanced, i.e. [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]≠Ø. Thus, an optimal and feasible schedule can be found by Algorithm 1 for the start-up process. This example belongs to Situation 2 of Algorithm 1. Therefore, the robot waiting time during the start-up process is set as follows: 1) During the process from M_(s0) to M_(s2), ω₀=0s and ω₁=90s; 2) During the process from M_(s2) to M_(s3), ω₀=ω₁=0 s and ω₂=79s; 3) During the process from M_(s3) to M_(s4), ω₀=ω₁=ω₂=0s and ω₃=65s; 4) During the process from M_(s4) to M_(s5), ω₀=ω₁=ω₂=ω₃=0s and ω₄=56s. In this way, an optimal and feasible schedule is obtained for the start-up process. The simulation result is shown in FIG. 4. It shows that it takes 521s for the start-up process.

In [Wu et al., 2008], a method is proposed to transfer the system to enter its steady state from the initial state. It puts a virtual token (wafer) in places p₂−p_(n) and none at p₁ in FIG. 2, i.e. M₀(p_(i))=1, i ∈N_(n)−{1}, and M₀(p₁)=0 at the initial state. Then, by running the PN model with a schedule obtained by the scheduling algorithm in [Wu et al., 2008], when all the virtual wafers go out of the system, the steady state is reached. It takes 563s for the start-up process. Therefore, by the method as disclosed herein in the present work, the time taken for the start-up process is reduced by 8.1% compared with the method in [Wu et al., 2008].

Example 2

The flow pattern is (PM₁, PM₂, PM₃, PM₄). α=5s, α₀=10s, μ=2s, a₁=85s, a₂=85s, a₃=110s, a₄=120s, and δ₁=δ₂=δ₃=δ₄=20s hold.

It follows from (5)-(8) that, we have

_(1L)=116s,

_(1U)=136s,

_(2L)=111s,

_(2U)=131s,

_(3L)=136s,

_(3U)=156s,

_(4L)=146s,

_(4U)=166s, and ψ₁=75s. By Theorem 2, the single-arm cluster tool is schedulable. For the steady state, an optimal and feasible schedule is obtained by setting ω₀=10s, ω₁=15s, ω₂=ω₃=0s, and ω₄=46s. Then, the cycle time of the system under the steady state is 146s. For this example, differences between the workloads among the steps are too large and [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Øholds. Thus, LPM is used to find an optimal and feasible schedule for the start-up process. With LPM, the robot waiting time during the start-up process is set as follows: 1) During the process from M_(s0) to M_(s2), ω₀=ω₀ ¹=10s and ω₁=ω₁ ¹=85s; 2) During the process from M_(s2) to M_(s3), ω₀=ω₀ ²=0s, ω₁=ω₁ ²=15s and ω₂=ω₂ ²=54s; 3) During the process from M_(s3) to M_(s4), ω₀=ω₀ ³=10s, ω₁=ω₁ ³=15s, ω₂=ω₂ ³=0s, and ω₃=ω₃ ³60s. Then, the tool enters its steady state and it is scheduled by setting ω₀=ω₀ ⁴=10s, ω₁=ω₁ ⁴=15s, ω₂=ω₃=ω₂ ⁴=ω₃ ⁴=0s, and ω₄=ω₄ ⁴=46s. In this way, an optimal and feasible schedule is obtained for the start-up process. The simulation result is shown in FIG. 5. It shows that it takes 405s for the start-up process. However, with the existing method in [Wu et al., 2008], it takes 455s for the start-up process. Thus, the time taken for this start-up process is reduced by 11%.

E. THE PRESENT INVENTION

The present invention is developed based on the theoretical development in Sections A-C above.

An aspect of the present invention is to provide a computer-implemented method for scheduling a cluster tool. The cluster tool comprises a single-arm robot for wafer handling, a LL for wafer cassette loading and unloading, and n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, i ∈N_(n), is used for performing Step i of the n wafer-processing steps for each wafer. Note that although the cluster tool is said to comprise a LL, it is understood that in the present invention, the cluster tool can have one or more LLs.

The method includes scheduling a start-up process for the cluster tool. The start-up process has plural system states M_(si), i=0, 1, . . . , n−1, where M_(s0) is an initial state of system start-up, and M_(si), 1≦i≦n−1 denotes that i instances of a wafer unloading from the robot to any one of the n process modules have occurred since the system start-up.

Advantageously, the start-up process is developed based on Scheduling Algorithm 1. When

_(max)≦

_(1U) and ψ₁≦

_(IU), i=1, 2, . . . , n, values of ω₀, ω₁, . . . , ω_(d) for each of the system states M_(sd), d=0, 1, . . . n−1, are determined. As mentioned above, ω_(j), j ∈ {0, 1, . . . , d}, is a robot waiting time used in the state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+l)th process module. According to Scheduling Algorithm 1, the values of ω₀, ω₁, . . . , and ω_(d) are determined by: setting ω₀=0 and ω₁=a₁ for the states M_(s0) and M_(s1); and setting ω_(i)=0, i ∈N_(d−1)∪{0}, and ω_(d)=max{

_(dmax)−ψ_(sd(d+1)1), 0} for the state M_(sd), 2≦d≦n−1 when n>2.

As a steady-state process follows the start-up process, preferably the method further includes scheduling the steady-state process based on the results obtained in the start-up process. In particular, values of ω₀, ω₁, . . . , and ω_(d) are determined, in which ω_(j), j ∈ {0, 1, . . . , n}, is a robot waiting time, used in a steady state of the cluster tool, for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module. As indicated in Section B.2 above, one option is to set ω_(i)=0, i ∈N_(n−1)∪{0}, and ω_(n)=max{

_(max)−ψ₁, 0}.

Also advantageously, the start-up process is further developed based on the LPM model. When [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, values of ω₀ ^(d), ω₁ ^(d), . . . , ω_(d) ^(d) for each of the system states M_(sd), d=0, 1, . . . , n−1, are determined, where ω_(j) ^(d), j ∈{0, 1, . . . d}, is a robot waiting time used in the state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module. The values of ω₀ ^(d), ω₁ ^(d), . . . , and ω_(d) ^(d), d=0, 1, . . . , n, are numerically optimized such that (18) is minimized subject to constraints (19)-(33). For the steady-state process, Theorem 5 indicates that one option is to set ω_(i)=ω_(i) ^(n), 0≦i≦n−1, and ω_(n)=

_(max)−Σ_(i=0) ^(n−1)ω_(i) for use in the state M_(sn) and thereafter. Similarly, M_(sn) denotes that n instances of a wafer unloading from the robot to any one of the n process modules have occurred since the system start-up.

The embodiments disclosed herein may be implemented using general purpose or specialized computing devices, computer processors, or electronic circuitries including but not limited to digital signal processors (DSP), application specific integrated circuits (ASIC), field programmable gate arrays (FPGA), and other programmable logic devices configured or programmed according to the teachings of the present disclosure. Computer instructions or software codes running in the general purpose or specialized computing devices, computer processors, or programmable logic devices can readily be prepared by practitioners skilled in the software or electronic art based on the teachings of the present disclosure.

In particular, the method disclosed herein can be implemented in a single-arm cluster tool if the cluster tool includes one or more processors. The one or more processors are configured to execute a process of scheduling the cluster tool according to one of the embodiments of the disclosed method.

The present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiment is therefore to be considered in all respects as illustrative and not restrictive. The scope of the invention is indicated by the appended claims rather than by the foregoing description, and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. 

1. A computer-implemented method for scheduling a cluster tool, the cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, i ∈ {1, 2, . . . , n}, is used for performing Step i of the n wafer-processing steps for each wafer, and a specialized processor configured to control the robot, the loadlock, and the process modules through execution of the method, the method comprising: when

_(max)≦

_(iU) and ψ₁≦

_(iU), i=1, 2, . . . , n, determining, by the specialized processor, values of ω₀, ω₁, . . . , ω_(d) for each of plural system states M_(sd), d=0, 1, . . . n−1, of the cluster tool, where M_(st), 1≦i≦n−1 denotes that i instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up, and ω_(j), j ∈{0, 1, . . . d}, is a robot waiting time used in the state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module; wherein the determining of ω₀, ω₁, . . . , ω_(d) for each of the system states M_(sd), d=0, 1, . . . , n−1, comprises: setting, by the specialized processor, ω₀=0 and ω₁=a₁ for the states M_(s0) and M_(s1)); where:

_(max)=max{

_(iL), i ∈ N_(n)}; ψ₁=2(n+1)μ+(2n+1 )α+α₀

_(iU) =a _(i)+4α+3μ+δ_(i) , i ∈ N _(n)\{1};

_(1U) =a _(i)+3α+α₀+3μ+δ₁;

_(1L) =a _(i)+4α+μ, i ∈ N _(n)\{1};

_(1L) =a ₁+3α+α₀+3μ; a_(i), i ∈ N_(n), is a time that a wafer is processed in the ith process module; δ_(i) is the wafer residency time constraint of Step i, given by a pre-determined longest time for which a wafer in the ith process module is allowed to stay therein after this wafer is processed; α is a time of loading a wafer to or unloading the wafer to the robot in Step i; μ is a time of the robot moving from one wafer-processing step to another; α₀ is a time of the robot unloading a wafer from the loadlock and aligning the same; and N_(m)={1 ,2, . . . ,} for a positive integer m.
 2. The method of claim 1, wherein the determining of ω₀, ω₁, . . . , ω_(d) for each of the system states M_(sd), d=0, 1, . . . , n−1, further comprises: setting ω_(i)=0, i ∈ N_(d−1)∪{0}, and ω_(d)=max{

_(dmax)−ψ_(sd(d+1)1), 0} for the state M_(sd), 2≦d≦n−1 when n>2; where:

_(dmax)=max{

_(iL), i ∈ N_(d)}; and ψ_(sd(d+1)1)=2(d+1)μ+(2d+1)α+α₀.
 3. The method of claim 2, further comprising: when

_(max)≦

_(iU) and ψ₁≦

_(iU), i=1, 2, . . . , n, setting, by the specialized processor, ω_(i)=0, i ∈ N_(n−1)∪{0}, and ω_(n)=max{

_(max)−ψ₁0} for use in a state M_(sn) and thereafter, where M_(sn) denotes that n instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up, and ω_(j), j ∈ {0, 1, . . . n}, is a robot waiting time, used in a steady state of the cluster tool, for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module.
 4. The method of claim 3, further comprising: when [

_(1L),

_(1U)], ∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]Ø, determining, by the specialized processor, values of ω₀ ^(d), ω₁ ^(d), . . . , ω_(d) ^(d) for each of the system states M_(sd), d=0,1, . . . , n−1, where ω_(j) ^(d), j ∈ {0, 1, . . . , d}, is a robot waiting time used in the state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module; wherein the determining of ω₀ ^(d), ω₁ ^(d), . . . , ω_(d) ^(d) for each of the system states M_(sd), d=0, 1, . . . , n, comprises: numerically optimizing the values of ω₀ ^(d), ω₁ ^(d), . . . , ω_(d) ^(d) such that $\sum\limits_{d = 1}^{n}\; {\sum\limits_{i = 0}^{d}\; \omega_{i}^{d}}$ is minimized subject to: t ₁₁ ⁰=α₀+μ+α, t ₁₂ ¹ =t ₁₁ ⁰+ω₁ ¹+α, t _(i1) ^(d) =t _((i=1)2) ⁰+μ+α, 1≦i≦d+1 and 1≦d≦n−1, t _(i2) ^(d) =t _((i+2)1) ^(d)+μ+ω_(i) ^(d)+α, 1≦i≦d−1 and 1≦d≦n−1, t ₀₂ ^(d) =t _((d+2)1) ^(d)+μ+ω₀ ^(d)+α₀, 1≦d≦n−1, t _(d2) ^(d) =t ₁₁ ^(d−1)+μ+ω_(d) ^(d)+α, 1≦d≦n−1, t _(i1) ^(d) =t _((i+1)2) ^(d)+μ+α, 1≦i≦n and d ∈ {n,n+1}, t ₀₁ ^(d) =t _(n2) ^(d) +μα, ∈ {n, n+1}, t _(i2) ^(d) =t _((i+2)1) ^(d)+μ+ω_(i) ^(d)+α, 1≦i≦n−2 and d ∈ {n, n+1}, t _((n−1)2) ^(d) =t ₀₁ ^(d)+μ+ω_(n−1) ^(d) +α, d ∈ {n, n+1}, t _(n2) ^(d) =t ₁₁ ^(d−1)+μ+ω_(n) ^(d) +α, d ∈ {n, n+1}, t ₀₂ ^(d) =t ₂₁ ^(d)+μ+ω₀ ^(d)+α₀ , d ∈ {n, n+1}, ω_(i) ^(n)=ω_(l) ^(n+1), 0≦i≦n ${{\sum\limits_{d = 1}^{n}\; {\sum\limits_{i = 0}^{d}\; \omega_{i}^{d}}} \geq 0};$ and a _(i) ≦t _(i2) ^(d) −α−t _(i1) ^(d−1) ≦a _(i)+δ_(i), 1≦i≦d and 1≦d≦n+1; where: t_(i1) ^(d) denotes a time when the robot completes loading a wafer into Step i, i ∈ N_(n)∪{0}; t_(i2) ^(d) a time when the robot completes unloading a wafer from Step i, i ∈ N_(n); t₀₂ ^(d) denotes a time when the robot completes unloading the loadlock and aligning the same; and ω_(j) ^(d), j ∈ N_(n) and d ∈ {n, n+1}, is a robot waiting time used in a state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module, the state M_(sd) denoting that d instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up.
 5. The method of claim 4, further comprising: when [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, by the specialized processor, ω_(i)=ω_(i) ^(n), 0≦i≦n−1, and ω_(n)=

_(max)−Σ_(t=0) ^(n−1)ω_(i) for use in the state M_(sn) and thereafter.
 6. A computer-implemented method for scheduling a cluster tool, the cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, i ∈ {1, 2, . . . , n}, is used for performing Step i of the n wafer-processing steps for each wafer, and a specialized processor configured to control the robot, the loadlock, and the process modules through execution of the method, the method comprising: when [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, determining, by the specialized processor, values of ω₀ ^(d), ω₁ ^(d), . . . , ω_(d) ^(d) for each of plural system states M_(sd), d=0,1, . . . , n−1, of the cluster tool, where M_(s1), 1≦i≦n−1 denotes that i instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up, and ω_(j) ^(d), j ∈ {0, 1, . . . d}, is a robot waiting time used in the state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module; wherein the determining of ω₀ ^(d), ω₁ ^(d), . . . ω_(d) ^(d) for each of the system states M_(sd), d=0, 1, . . . , n, comprises: numerically optimizing the values of ω₀ ^(d), ω₁ ^(d), . . . , ω_(d) ^(d) such that $\sum\limits_{d = 1}^{n}\; {\sum\limits_{i = 0}^{d}\; \omega_{i}^{d}}$ is minimized subject to: t ₁₁ ⁰=α₀+μ+α, t ₁₂ ¹ =t ₁₁ ⁰+ω₁ ¹+α, t _(i1) ^(d) =t _((i=1)2) ⁰+μ+α, 1≦i≦d+1 and 1≦d≦n−1, t _(i2) ^(d) =t _((i+2)1) ^(d)+μ+ω_(i) ^(d)+α, 1≦i≦d−1 and 1≦d≦n−1, t ₀₂ ^(d) =t _((d+2)1) ^(d)+μ+ω₀ ^(d)+α₀, 1≦d≦n−1, t _(d2) ^(d) =t ₁₁ ^(d−1)+μ+ω_(d) ^(d)+α, 1≦d≦n−1, t _(i1) ^(d) =t _((i+1)2) ^(d)+μ+α, 1≦i≦n and d ∈ {n,n+1}, t ₀₁ ^(d) =t _(n2) ^(d) +μα, ∈ {n, n+1}, t _(i2) ^(d) =t _((i+2)1) ^(d)+μ+ω_(i) ^(d)+α, 1≦i≦n−2 and d ∈ {n, n+1}, t _((n−1)2) ^(d) =t ₀₁ ^(d)+μ+ω_(n−1) ^(d) +α, d ∈ {n, n+1}, t _(n2) ^(d) =t ₁₁ ^(d−1)+μ+ω_(n) ^(d) +α, d ∈ {n, n+1}, t ₀₂ ^(d) =t ₂₁ ^(d)+μ+ω₀ ^(d)+α₀ , d ∈ {n, n+1}, ω_(i) ^(n)=ω_(l) ^(n+1), 0≦i≦n ${{\sum\limits_{d = 1}^{n}\; {\sum\limits_{i = 0}^{d}\; \omega_{i}^{d}}} \geq 0};$ and a _(i) ≦t _(i2) ^(d) −α−t _(i1) ^(d−1) ≦a _(i)+δ_(i), 1≦i≦d and 1≦d≦n+1. where:

_(iU) =a _(i)+4α+3μ+δ_(i) , i ∈ N _(n)\{1};

_(1U) =a ₁+3α+α₀+3μ+δ₁;

_(iL)=a_(i)+4α+3μ, i ∈N_(n)\{1};

_(1L)=a₁+3α+α₀+3μ; t_(i1) ^(d) denotes a time when the robot completes loading a wafer into Step i, i ∈ N_(n)∪{0}; t_(i2) ^(d) denotes a time when the robot completes unloading a wafer from Step i, i ∈ N_(n); t₀₂ ^(d) denotes a time when the robot completes unloading the loadlock and aligning the same; ω_(j) ^(d), j ∈ N_(n) and d ∈ {n, n+1}, is a robot waiting time used in a state M_(sd) for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module, the state M_(sd) denoting that d instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up; a_(i), i ∈ N_(n), is a time that a wafer is processed in the ith process module; δ_(i) is the wafer residency time constraint of Step i, given by a pre-determined longest time for which a wafer in the ith process module is allowed to stay therein after this wafer is processed; α is a time of loading a wafer to or unloading the wafer to the robot in Step i; μ is a time of the robot moving from one wafer-processing step to another; α₀ is a time of the robot unloading a wafer from the loadlock and aligning the same; and N_(m)={1, 2, . . . , m} for a positive integer m.
 7. The method of claim 6, further comprising: when [

_(1L),

_(1U)]∩[

_(2L),

_(2U)]∩ . . . ∩[

_(nL),

_(nU)]=Ø, setting, by the processor, ω_(i)=ω_(i) ^(n), 0≦i≦n−1, and ω_(n)=

_(max)−Σ_(i=0) ^(n−1)ω_(i) for use in the state M_(sn) and thereafter, where ω_(j), j ∈ {0, 1, . . . n}, is a robot waiting time, used in a steady state of the cluster tool, for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module.
 8. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 1. 9. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 2. 10. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 3. 11. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 4. 12. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 5. 13. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 6. 14. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim
 7. 